Tí ulo a ículo / Tí ol a icle:
Hippocampal shape analysis in Alzheime ’s
disease using Func ional Da a Analysis
Au o es / Au o s
I ene Epi anio, Noelia Ven u a-Campos
Re is a:
S a is ics in Medicine (2013)
Ve sión / Ve sió:
P ep in del au o
Ci a bibliog á ica / Ci a
bibliog à ica (ISO 690):
EPIFANIO, I ene; VENTURA-CAMPOS, Noelia.
Hippocampal shape analysis in Alzheime ’s
disease using Func ional Da a Analysis.
S a is ics in Medicine, 2013.
u l Reposi o i UJI:
h p://hdl.handle.ne /10234/73306
Resea ch A icle S a is ics
in Medicine
Recei ed XXXX
(www.in e science.wiley.com) DOI: 10.1002/sim.0000
Hippocampal shape analysis in Alzheime ’s
disease using Func ional Da a Analysis
I ene Epi anioa∗, Noelia Ven u a-Camposb
The hippocampus is one o he i s a ec ed egions in Alzheime ’s disease. Le hippocampi o con ols, mild
cogni i e impai men and Alzheime ’s disease pa ien s a e ep esen ed by sphe ical ha monics. Func ional da a
analysis is used in he hippocampal shape analysis. Func ional p incipal componen analysis and uncional
independen componen analysis a e de ined o mul i a ia e unc ions wi h wo a gumen s. A unc ional linea
disc iminan unc ion is also de ined.Compa isons wi h o he app oaches a e ca ied ou . Ou unc ional app oach
gi es p omising esul s, especially in shape classi ica ion. Copy igh c
0000 John Wiley & Sons, L d.
Keywo ds: Func ional da a analysis; Shape analysis; Alzheime ’s disease; P incipal componen analysis;
Independen componen analysis; Disc iminan analysis
1. In oduc ion
The ea ly diagnosis o Alzheime ’s disease (AD) is a c ucial issue in ou socie y, because he adminis a ion o medicines
o indi iduals who a e sub ly impai ed may ende he ea men s mo e e ec i e. Mild cogni i e impai men (MCI) is
conside ed as a diagnos ic en i y wi hin he con inuum o cogni i e decline owa ds AD in old age [1,2]. Longi udinal
s udies show a di ec ela ion be ween he hippocampal olume dec ease and cogni i e decline [3,4]. Howe e ,
olume ic measu emen s a e simplis ic ea u es and s uc u al changes a speci ic loca ions canno be e lec ed in hem.
I mo phological changes could be es ablished, hen his should enable esea che s o gain an inc eased unde s anding o
he condi ion. This is he eason why nowadays shape analysis is o a g ea impo ance in neu oimaging [5].
Se e al shape modeling app oaches ha e been conside ed in he neu oimaging li e a u e. One o hem is he medial
ep esen a ion, whe e he bina y objec is ep esen ed using a se o a oms and links ha connec he a oms oge he o
o m a skele al ep esen a ion o he objec . S yne e al. [6] applied his scheme o hippocampi and o he human b ain
s uc u es. The dis ance map app oach has been applied in classi ying a collec ion o hippocampi in Golland e al. [7]. In
a dis ance map, he dis ance om each poin in he image o he bounda y o he objec is compu ed. O he app oaches
include de o ma ion ields ob ained by wa ping indi idual subs uc u es o a empla e, such as he pape o Joshi e al. [8]
applied o he hippocampus, o he landma k app oach used wi h hippocampi by Pa k e al. [9] o Shen e al. [10]. Ins ead
o he p e ious non-pa ame ic models, a pa ame ic app oach, which has been success ully applied o model a ious
subco ical s uc u es, is he sphe ical ha monic ep esen a ion (SPHARM) [11,12,13,14,15,16,17]. Each indi idual
su ace is pa ame e ized by a se o coe icien s weigh ing he basis unc ions: he sphe ical ha monics o i s weigh ed
e sion ( he weigh ed sphe ical ha monic ep esen a ion). S yne e al. [5] compa ed he sampled bounda y implied by he
SPHARM desc ip ion wi h he medial shape desc ip ion, ob aining good conco dance be ween bo h desc ip ions. O he
wo ks p opose global ea u es which disc imina e he condi ion [18]. I is qui e common o use some o hese app oaches
wi h a p incipal componen analysis o shape classi ica ion and g oup compa ison.
The sphe ical ha monic ep esen a ion is a pa icula case o ep esen ing unc ional da a as smoo h unc ions. The
whole su ace is modeled om a se o poin s belonging o he su ace. Func ional da a a e obse ed disc e ely al hough
a con inuous unc ion lies behind hese da a. In o de o con e he disc e e obse a ions in o a ue unc ional o m, each
aDep . Ma em`a iques, Uni e si a Jaume I, Campus del Riu Sec, 12071 Cas ell´o, Spain
bDep . Psicologia B`asica, Cl´ınica i Psicobiologia, Uni e si a Jaume I, Spain
∗Co espondence o: Tel.: +34-964728390, ax: +34-964728429. E-mail: epi [email protected]
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unc ion is app oxima ed (smoo hed) by a weigh ed sum (a linea combina ion) o known basis unc ions. Func ional da a
analysis (FDA) p o ides s a is ical p ocedu es o unc ional obse a ions (a whole unc ion is a da um). The goals o FDA
a e basically he same as hose o any o he b anch o s a is ics. Ramsay and Sil e man [19] gi e an excellen o e iew.
Fe a y and Vieu [20] p o ide a complemen a y and e y in e es ing iew on nonpa ame ic me hods o unc ional da a.
A mix u eo p ac ical and heo e ical aspec s is ound in Fe a y and Romain [21]. The ield o FDA is qui e new and he e
is s ill a lo o wo k o be done, bu in ecen yea s se e al applica ions ha e been de eloped in di e en ields, especially
in human heal h [22,23,24].
In Epi anio and Ven u a-Campos [25] wo-dimensional (2D) shapes we e analyzed om he h ee poin o iews
conside ed by S oyan and S oyan [26] o desc ibing shapes: i s ly, se desc ip o s; secondly, using landma ks (poin
desc ip ion); and hi dly, employing a unc ion desc ibing he con ou s. The esul s we e compa ed wi h hese h ee
app oaches ( he se heo y app oach, he landma k based app oach and he unc ional app oach) in wo o he main
p oblems in o m s a is ics: he s udy o he main sou ces o a ia ion among he shapes (p incipal componen analysis,
PCA), and classi ica ion among di e en classes (disc iminan analysis). The analysis o con ou unc ions by FDA ga e
mo e meaning ul esul s in bo h p oblems.
In his wo k, he hippocampus su ace is desc ibed by mul i a ia e ( h ee) unc ions wi h wo a gumen s. In Sec ion 2,
he me hodologyis in oduced oge he wi h ou da a. Wediscuss he ex ensiono he PCA o deal wi h i a ia e unc ional
da a wi h wo a gumen s. A disc iminan unc ion based on independen componen analysis (ICA) is de ined o indica ing
whe e he di e ences be ween g oups a e and wha hei le el o disc imina ionis. In Sec ion 3, he me hodology is applied
o he analysis o s uc u al magne ic esonance imaging(sMRI) scans o s udying he hippocampaldi e ences among he
subjec s o h ee g oups: cogni i ely no mal (CN) subjec s, pa ien s wi h mild cogni i e impai men (MCI), and pa ien s
wi h ea ly Alzheime ’s disease (AD). Compa ison wi h o he wo ks is ca ied ou . Finally, conclusions and some open
p oblems a e discussed in Sec ion 4.
2. Ma e ials and me hods
2.1. B ain sMRI scans p ocessing
A o al o 28 indi iduals (12 CN, 6 MCI and 10 AD subjec s) a e analyzed in his s udy, whose desc ip ion is in Table
1. All he indi iduals we e ec ui ed om he Neu ology Se ice a La Magdalena Hospi al (Cas ell´o, Spain) and he
Neu opsychology Se ice a he Uni e si a Jaume I. All expe imen al p ocedu es complied wi h he guidelines o he
e hical esea ch commi ee a he Uni e si a Jaume I. W i en in o med consen was ob ained om e e y subjec o
hei app op ia e p oxy p io o pa icipa ion. Selec ion o he pa icipan g oup was made a e ca e ul neu ological and
neu opsychological assessmen . The neu opsychological es ba e y in ol ed Digi Span, Simila i ies, Vocabula y, and
Block Design o he WAIS-III; Lu ia’s Wa ches es , and Poppel eu e s O e lapping Figu e es . sMRI we e acqui ed on a
1.5T scanne (Gene al Elec ic). A whole b ain high esolu ion T1-weigh ed ana omical e e ence scan was acqui ed (TE
4.2 ms, TR 11.3 ms, FOV 24 cm; ma ix = 256×256×124, 1.4 mm- hick co onal images).
Hippocampi we e aced manually on con iguous co onal slices (o sec ions) ollowing he guidelines o Wa son e al.
[27], and Hasboun e al. [28]. The hippocampus segmen a ion was done by an expe ace wi h MRic o so wa e, blinded
o he clinical da a o he s udy subjec s. The segmen a ion o each hippocampus las ed app oxima ely 40 minu es. An
example o he le and igh hippocampal con ou (d awn in whi e) in a co onal iew is shown in Figu e 1(a), while
a sagi al iew o one o he hippocampus can be seen in Figu e 1(b). Images we e isually eo ien ed. The an e io
commissu e–pos e io commissu e (ACPC) line was iden i ied and he images we e hen eo ien ed pa allel o i . The
slices we e pu oge he using he isosu ace unc ion in Ma lab, which gi es he e ices and aces o he iangle mesh.
2.2. Su ace pa ame iza ion
In 2D, he con ou is a closed plana cu e ha consis s o he elemen s o he igu e bounda y. The con ou pa ame iza ion
by i s a c leng h can be applied o any con ou (no e ha o he con ou unc ions ha e limi a ions, see Kind a enko [29]
o a e iew o a ious con ou unc ions).
This me hod has been ex ended o ep esen analogously he su aces o closed 3D objec s. In his case, ins ead o wo
pa ame ic unc ions wi h one pa ame e , h ee unc ions wi h wo angula pa ame e s a e needed: x(θ, φ),y(θ, φ),z(θ, φ)
(see L. Shen and McPeek [15] o a de ailed explana ion). Speci ically, a su ace is mapped on o a uni sphe e unde a
bijec i e mapping. Howe e , unlike he 2D case, some p ac ical p oblems p e en his mapping om being comple ely
s aigh o wa d. In ac , his one- o-one mapping can be ob ained om a ious su ace la ening echniques such as
con o mal mapping [12], semi-isome ic mapping [30], a ea p ese ing mapping [13,31] and he de o mable su ace
algo i hm [32]. Since he con o mal mapping ends o in oduce huge a ea dis o ion, a ea p ese ing mapping is widely
used. Howe e , hese la ening me hods a e compu a ionally in ensi e and no i ial o implemen . He e, we use a new
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al e na i e p oposed ecen ly in Chung e al. [17] o objec s ha a e close o ei he s a -shape o con ex. The mapping
is based on he equilib ium s a e o hea di usion. The idea is acing he geodesic pa h o hea equilib ium s a e om
a hea sou ce (hippocampus in his case) o a hea sink (sphe e). As sol ing an iso opic hea equa ion in a 3D image
olume is compu a ionally i ial, his la ening echnique is nume ically simple han any o he a ailable me hods and
does no equi e op imizing a cos unc ion. De ails abou his me hod can be ound in Chung e al. [17], and how i wo ks
can be seen a h p://www.s a .wisc.edu/∼mchung/ esea ch/amygdala/. This s ep al hough necessa y is auxilia , and any
o he me hod can be used wi hou changing he ollowing analysis. Howe e , he me hod om Chung e al. [17] has been
e ec i e wi h hippocampi, and we will use i .
Once he su ace is mapped on o hesphe e, he anglesse e as coo dina es o ep esen ing hippocampussu aces using
basis unc ions. Figu e 2shows an illus a ion o he su ace la ening p ocess o one hippocampus wi h ha me hod, and
he su ace pa ame e iza ion using he angles (θ, ϕ). The poin θ= 0 co esponds o he no h pole o a uni sphe e.
2.3. Rep esen ing unc ions by basis unc ions
The i s s ep in FDA is he con e sion om disc e e da a o unc ions by smoo hing. Linea combina ions o basis
unc ions a e used o ep esen ing unc ions. We ha e chosen as basis he sphe ical ha monics, because hey ha e been
al eady used in simila s uc u es wi h excellen esul s, u he mo e i s o hogonali yhas compu a ionalad an ages. O he
possible basis could be he weigh ed Fou ie se ies [14], sphe ical splines [33,34] o sphe ical wa ele s [35,36], al hough
sphe ical ha monics is he mos used basis in his ield.
Al houghcomplex- aluedsphe ical ha monics could be used as in Ge ig e al. [11] o Shen e al. [13], we ha e p e e ed
o used eal sphe ical ha monics as in Chung e al. [14,17], conside ing ha mos applica ions o sphe ical ha monics
equi e only eal- alued sphe ical unc ions, and o con enience in se ing up a eal- alued s ochas ic model.
A eal basis o sphe ical ha monics is gi en by (lis he deg ee and mis he o de ):
Ylm(θ, ϕ) = 


√2N(l,m)cos(mϕ)Pm
l(cosθ)i m > 0
N(l,0)P0
l(cosθ)i m=0
√2N(l,|m|)sin(|m|ϕ)P|m|
l(cosθ)i m < 0
(1)
whe e N(l,m)=q2l+1
4π
(l−m)!
(l+m)! and Pm
lis he associa ed Legend e polynomial o o de mde ined o e he ange [−1,1]:
Pm
l(x) = (−1)m
2ll!(1 −x2)m/2dl+m
dxl+m(x2−1)l.
Le S2be he uni sphe e in R3, and and g∈L2(S2). The inne p oduc is de ined by
< , g >=Zπ
θ=0 Z2π
ϕ=0
(θ, ϕ)g(θ, ϕ)dΩ = ZS2
(θ, ϕ)g(θ, ϕ)dΩ = ZS2
gdΩ(2)
whe e dΩ=sin(θ)dϕdθ. Wi h espec o he inne p oduc , he sphe ical ha monics sa is y he o hono mal condi ion:
RS2YlmYl′m′dΩ = δll′δmm′, whe e δij is he K oneke ’s del a.
The h ee unc ions a e independen ly exp essed in e ms o he sphe ical ha monic as: x(θ, ϕ)=
PL
l=0 Pl
m=−lcx
lmYlm(θ, ϕ),y(θ, ϕ)=PL
l=0 Pl
m=−lcy
lmYlm(θ, ϕ)and z(θ, ϕ)=PL
l=0 Pl
m=−lcz
lmYlm(θ, ϕ).Lis he
maximal deg ee o he ep esen a ion, which de e mines he deg ee o which da a a e smoo hed. As we know he alues o
each unc ion in a sample o poin s {(θi, ϕi)}n
i=1, he coe icien s can be es ima ed by leas squa es. In he case o x(θ, ϕ)
(analogously o he o he wo unc ions), le x={x(θi, ϕi)}n
i=1 be he ec o o obse a ions, cx he ec o con aining
he coe icien s cx
lm and Y={Ylm(θi, ϕi)}n
i=1 he ma ix o basis unc ion alues a he obse a ion poin s, hen cx=
(Y′Y)−1Y′x. I he size o he linea equa ion is ex emely la ge, he coe icien s can be also es ima ed in a leas squa es
ashion by he i e a i e esidual i ing (IRF) algo i hm [14]. Finally, a ec o - alued unc ion can be buil F(θ, ϕ)=
(x(θ, ϕ), y(θ, ϕ), z(θ, ϕ))′=PL
l=0 Pl
m=−lclmYlm(θ, ϕ), whe e clm =(cx
lm, cy
lm, cz
lm)′.
As in o he wo ks ha used SPHARM o simple su aces, a small L esul s in an accep able deg ee o smoo hing o
his kind o subco ical s uc u es. Fo mo e complex s uc u es, such as co ical su aces, a highe deg ee ( o example,
52) is necessa y o a good ep esen a ion [14]. A e inspec ing he ep esen a ions o di e en alues o L,L= 15 was
isually chosen o ou hippocampi. Figu e 2shows he sphe ical ha monic ep esen a ions o a hippocampus su ace o
di e en L. No e ha i Lis oo small, we miss impo an aspec s o he su ace, bu wi h la ge Lwe no only i da a bu
also noise. Hence he e was an ine i able ade-o be ween hese wo ac o s in choosing L= 15. In o de o check i he
analysis is obus o he choice o di e en alues o L, he classi ica ion esul s o di e en alues o La e shown in he
supplemen a y ma e ial.
Some imes, a egis a ion o alignmen is hen ca ied ou . Howe e , in his case i is no necessa y as he loca ion was
emo ed p e iously by ansla ing each hippocampus o he same poin in such a way ha i s cen oid coincides wi h ha
poin ((25,25,25) in ou case). No e ha he cen e o he sphe e o he su ace pa ame iza ion is on each hippocampal
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cen oid, as in Chung e al. [17]. Fu he mo e, all he hippocampi ha e he same o ien a ion, so no o a ion is needed.
As size in o ma ion (hippocampal olume is a usual disc imina o y ea u e) is impo an , no scaling co ec ion is ca ied
ou . So we analyze he o m, which combines he shape and he size in o ma ion, o he wise scale can be emo ed by
di iding h ough he cen oid’s size a he beginning, as in Epi anio and Ven u a-Campos [25]. He e, no u he alignmen
is necessa y, as in Chung e al. [17], since he coo dina es ((θ, ϕ)) on wo su aces a e co esponding pai s, and he e o e
he coe icien s ma ch each o he . Fo o he kind o he su ace pa ame iza ion, an alignmen could be necessa y as
explained in L. Shen and McPeek [15], whe e landma ks a e used o egis a ion.
In his pape he a gumen s a e angles, bu when he a gumen is ime, unc ions usually exhibi wo kind o a ia ion:
ampli ude and phase a ia ion. The is one accoun s o he size o he shape ea u es in he unc ions, whe eas he second
one e e s o he loca ion o he ea u es. In case ha we had phase a ia ion, he algo i hm combining egis a ion wi h
p incipal componen s analysis in Kneip and Ramsay [37] could be used, whe e decomposi ion o unc ional a ia ion in o
ampli ude and phase pa i ions is de ined.
2.4. Func ional disc iminan analysis
Linea disc iminan analysis can be used wi h unc ions, bu egula iza ion is necessa y o gi e meaning ul esul s [22,
ch. 8]. Recen ad ances in unc ional da a classi ica ion ha e been epo ed by a ious au ho s. Many o hem in ol e
a ype o p ep ocessing (some imes implici ) o he unc ional da a (see [38] o a compa ison o di e en me hods o
uni a ia e unc ions, and [25] o mul i a ia e unc ions wi h one a gumen ). One possible egula iza ion app oach is o
concen a e on he i s ew p incipal componen s as in [22, ch. 8], o some o he ini e-dimensional ep esen a ion o
he da a, as ICA, which has gi en be e esul s han PCA and o he al e na i es in p e ious li e a u e [25]. So, once he
hippocampi a e ep esen ed in a basis (SPHARM), we can ca y ou he unc ional da a analysis, beginning wi h explo ing
he hippocampal a iabili y by PCA and ICA, and using hem o he disc iminan analysis.
2.5. Func ional PCA (FPCA)
Fo s udying he main sou ces o a ia ion among he hippocampi, p incipal componen analysis is used. In o de o see
how PCA wo ks in he unc ional con ex , le us ecall PCA o Mul i a ia e Da a Analysis (MDA). Sho ly, summa ions
change in o in eg a ions. In MDA, p incipal componen s a e ob ained by sol ing he eigenequa ion
Vξ=ρξ, (3)
whe e Vis he sample a iance-co a iance ma ix, V=(N−1)−1X′X,Xis he cen e ed da a ma ix, Nis he numbe
o subjec s obse ed, and X′indica es he anspose o X. Mo eo e , ρand ξa e an eigen alue and an eigen ec o o V,
espec i ely.
In he unc ional e sion o PCA, PCs a e eplaced by unc ions. Be o e analyzing ou mul i a ia e unc ional da a
wi h mul iple a gumen s, le us in oduce he unc ional uni a ia e case. Le {x1( ), . . . , xN( )}be he se o uni a ia e
obse ed unc ions wi h one scala a gumen . The mean unc ion is de ined as he a e age o he unc ions poin -wise
ac oss eplica ions (¯x( ) = N−1PN
i=1 xi( )). I da a ha e been cen e ed ( he mean unc ion has been sub ac ed), he
co a iance unc ion (s, ) is de ined analogously by (s, ) = (N−1)−1PN
i=1 xi(s)xi( ). As explained in Ramsay and
Sil e man [19, Chap e 8], he unc ional coun e pa o equa ion 3is he ollowing unc ional eigenequa ion
Z (s, )ξ( )d =ρξ(s),(4)
whe e ρis s ill an eigen alue, bu whe e ξ(s) is an eigen unc ion o he a iance-co a iance unc ion, a he han an
eigen ec o . Now, he p incipal componen sco e co esponding o ξ(s) is compu ed by using he inne p oduc o
unc ions: si=Rxi(s)ξ(s)ds. No e ha o mul i a ia e da a, he index sis no con inuous, bu a disc e e index j eplaces
i : si=Pjxijξj.
Fo sol ing he eigenequa ion4, he o iginal unc ions could be disc e ized. Howe e ,we will wo k wi h he coe icien s
o he unc ions exp essed as a linea combina ion o known basis unc ions. I he basis is o hono mal, FPCA educes
o he s anda d mul i a ia e PCA o he coe icien a ay, as explained in Ramsay and Sil e man [19, Sec. 8.4.2], whe e
compu a ional me hods o FPCA a e e iewed. This educes he amoun o in o ma ion gene a ed.
Wi h ega d o he numbe o PCs ha can be compu ed, le us no e ha in he unc ional con ex , “ a iables” now
co espond o alues o , and he e is no limi o hese. The e o e, a maximum o N – 1 componen s can be compu ed.
Howe e , i he numbe o basis unc ions M ep esen ing he unc ions is less han N,Mwould be he maximum.
2.5.1. FPCA wi h mul iple unc ions and mul iple a gumen s Le {Fi(θ, ϕ)}N
i=1 be he se o obse ed
unc ions. Each Ficonsis s o h ee unc ional da a wi h wo a gumen s ep esen ing one hippocampus
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((xi(θ, ϕ), yi(θ, ϕ), zi(θ, ϕ)). Th ee mean unc ions (¯x(θ, ϕ),¯y(θ, ϕ),¯z(θ, ϕ)) and h ee co a iance unc ions
( XX ((θ, ϕ),(ϑ, φ)), Y Y ((θ, ϕ),(ϑ, φ)), ZZ ((θ, ϕ),(ϑ, φ))) can be compu ed poin wisely as be o e o each kind
o unc ion, espec i ely. We can calcula e he c oss-co a iance unc ion o he cen e ed da a by (analogously o he
combina ion XZ and Y Z) XY ((ϑ, φ),(θ, ϕ)) = (N−1)−1PN
i=1 xi(ϑ, φ)yi(θ, ϕ).
An inne p oduc on he space o ec o - alued unc ions is de ined by summing he inne p oduc s o he componen s
(de ined in 2) as
< F1, F2>=< x1, x2>+< y1, y2>+< z1, z2> . (5)
A ypical PC is de ined by a h ee- ec o ξ=(ξX,ξY,ξZ) o weigh unc ions. Now, he PC sco e o he i- h unc ion
is compu ed by si=< Fi, ξ >=RS2xiξXdΩ + RS2yiξYdΩ + RS2ziξZdΩ. PCs a e solu ions o he eigenequa ion sys em
V ξ =ρξ, which in his case can be w i en as
RS2 XX ((ϑ, φ),(θ, ϕ))ξX(θ, ϕ)dΩ + RS2 XY ((ϑ, φ),(θ, ϕ))ξY(θ, ϕ)dΩ + RS2 XZ ((ϑ, φ),(θ, ϕ))ξZ(θ, ϕ)dΩ = ρξX(ϑ, φ)
RS2 Y X ((ϑ, φ),(θ, ϕ))ξX(θ, ϕ)dΩ + RS2 Y Y ((ϑ, φ),(θ, ϕ))ξY(θ, ϕ)dΩ + RS2 Y Z ((ϑ, φ),(θ, ϕ))ξZ(θ, ϕ)dΩ = ρξY(ϑ, φ)
RS2 ZX ((ϑ, φ),(θ, ϕ))ξX(θ, ϕ)dΩ + RS2 ZY ((ϑ, φ),(θ, ϕ))ξY(θ, ϕ)dΩ + RS2 ZZ ((ϑ, φ),(θ, ϕ))ξZ(θ, ϕ)dΩ = ρξZ(ϑ, φ).
(6)
To sol e he eigenequa ion sys em, each unc ion in he ec o - unc ion Fiis eplaced by a ec o o basis coe icien s,
and a single ec o is buil by joining hem oge he . Then, i ci= ({cx
ilm},{cy
ilm},{cz
ilm}) is ha ec o o coe icien s o
Fi, wi h l= 0, ..., Land m=−l o l, a ma ix Cwi h N ows (one pe indi idual) can be buil s acking hose ec o s. As
sphe ical ha monics a e o hono mal,we only need o compu e he PCA o C. When PCs ha e been compu ed, we sepa a e
he pa s belonging o each coo dina e, as explained in Ramsay and Sil e man [19, Sec. 8.5.1] o bi a ia e FPCA wi h
one a gumen . No e ha he compu a ion is educed when we wo k wi h he coe icien s ins ead o using a lo o a iables
ob ained by disc e izing he o iginal unc ions in a ine g id. Wi h L= 15 we ha e 256 coe icien s pe coo dina e, a o al
o 756 coe icien s ( a iables). Howe e , o a g id o 2562 e ices on he sphe e, he numbe o a iables ises o 7686.
Each eigen alue ρdi ided by he sum o all eigen alues gi es he p opo ion o a iance explained by each
eigen unc ion, as in he mul i a ia e case. Fu he mo e, o he j- h p incipal componen ξj= (ξj
X,ξj
Y,ξj
Z), he a ia ion
accoun ed o each coo dina e can be compu ed by < ξj
X, ξj
X>,< ξj
Y, ξj
Y>and < ξj
Z, ξj
Z> espec i ely, because hei
sum is one by de ini ion.
2.6. Func ional ICA (FICA)
ICA was success ully used o he classi ica ion o uni a ia e unc ions in Epi anio [38], whe e was compa ed wi h
classical and he mos ecen ad ances in uni a ia e unc ional da a classi ica ion gi ing esul s be e han o simila
o hose ob ained using he p e ious echniques in h ee di e en p oblems. Conc e ely, he p oposed desc ip o s we e
compa ed wi h he me hodology in oduced in Has ie e al. [39], in Fe a y and Vieu [40] including he mul i a ia e pa ial
leas -squa es eg ession (MPLSR) me hod in i s semi-me ic and PCA, in Rossi and Conan-Guez [41], in Fe ´e and Villa
[42], in Rossi and Villa [43], and as Li and Yu [44] use he same example, we can also compa e he esul s in Epi anio
[38] wi h hose o Li and Yu [44] (see Epi anio [38] o de ails). Tha me hodology was ex ended o he mul i a ia e case
wi h one a gumen in Epi anio and Ven u a-Campos [25], whe e he bes disc iminan esul s we e ob ained wi h he ICA
coe icien s compa ed wi h FPCA and o he al e na i es ( he penalized disc iminan analysis p oposed by Has ie e al.
[39] and he nonpa ame ic cu e disc imina ion me hod wi h he semi-me ic based on FPCA and MPLSR in oduced by
Fe a y and Vieu [40]) in he unc ional app oach, and he se and landma k app oach. He e we ex end he me hodology
o mul i a ia e unc ions wi h wo (o mo e) a gumen s, as i ICA has been use ul wi h he classi ica ion o unc ions wi h
one a gumen , he same can happen wi h mo e gene al unc ions.
Le us ecall ICA o MDA. Assume ha he da a ma ix Xis a linea combina ion o non-Gaussian (independen )
componen s i.e. X=SA whe e columns o Scon ain he independen componen s and Ais a linea mixing ma ix. ICA
a emp s o “un-mix” he da a by es ima ing an un-mixing ma ix Wwi h XW =S. Unde his gene a i e model, he
measu ed “signals” in Xwill end o be “mo e Gaussian” han he sou ce componen s (in S) due o he Cen al Limi
Theo em. Thus, in o de o ex ac he independen componen s o sou ces, we sea ch o an un-mixing ma ix W ha
maximizes he non-gaussiani y o he sou ces.
Fo uni a ia e unc ions, assume ha we obse e Nlinea mix u es x1( ), ..., xN( )o Kindependen componen s
sj( ):xi( ) = PK
j=1 aijsj( ), o all i. Each pai sj( )and sk( ), a each ime ins an , a e s a is ically independen . In
p ac ice, we ha e disc e ized unc ions (xi={xi( k); k= 1, ..., p}), he e o e we can conside he p×Nda a ma ix X=
{xi( k)}.
Howe e , unlike ou p e ious wo ks wi h one a gumen , ins ead o disc e izing he unc ions we will wo k wi h he
coe icien s in a unc ional basis o educing he compu a ional bu den. Suppose ha each unc ion has basis expansion:
xi( )=PM
m=1 bimGm( ). I we de ine xa ec o - alued unc ion wi h componen s x1, ..., xN, and G he ec o - alued
unc ion wi h componen s G1, ..., GM, we can exp ess he simul aneous expansion o all N unc ions as: x=BG, whe e
Bis he coe icien ma ix, wi h size N×M. I we pe o m ICA on B′, we ob ain B′=SbAb, so we can conside x
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=BG=A′
bS′
bG, i.e. he obse ed da a xa e gene a ed by a p ocess o mixing he Kcomponen s I=S′
bG( ows o
S′
bcon ain he independen componen s). The expansion o any unc ion ˜x( )no included in he o iginal xin e ms o
hese ICA componen s will be o he o m: ˜x( )=PK
j=1 ˜ajIj( ), wi h Ij( ) he j- h componen o I. I we es ima e I
and Gin ppoin s ({ k;k= 1, ..., p}), we can build he p×Kma ix Iand he p×Mma ix G, and hence I=GSb.
The K- ec o ˜a con aining he coe icien s ˜ajcan be easily ob ained by leas squa es i ing [19]: ˜a =(I′I)−1I′˜x, whe e
˜x ={˜x( k)}p
k=1. This yields ˜a =(S′
bG′GSb)−1S′
bG′˜x. Analogously, o he Gbasis, ˜x( )=PM
m=1 ˜
bmGm( ), and he
M- ec o ˜
bcon aining he coe icien s ˜
bmcan be compu ed as: ˜
b=(G′G)−1G′˜x. When he basis Gis o hono mal,
meaning ha G′Gis he iden i y ma ix :
˜a = (S′
bG′GSb)−1S′
bG′˜x = (S′
bSb)−1S′
b
˜
b.(7)
I he unc ions ha e mo e han one a gumen , he discussion is iden ical. When ha ing mul i a ia e unc ional da a, we
can conca ena e he coe icien s o each unc ion in o a single long ec o , as done in Sec. 2.5.1 o compu ing mul i a ia e
FPCA. In ou case, ˜
bwould be (ci)′.
Be o e he applica ion o he ICA algo i hm, i is use ul o educe he dimension o he da a p e iously by PCA ( o
de ails, see Hy ¨a inen e al. [45, Sec ion 5]), hus educing noise and p e en ing o e lea ning [46, Sec ion 13.2]. The e o e
we compu e he PCA i s , e aining a ce ain numbe o componen s, and hen es ima e he same numbe o independen
componen s as he PCA educed dimension.
2.6.1. Func ional linea disc iminan Func ional linea disc iminan can be used i he objec i e is also o disc imina e
be ween di e en g oups and o unde s and he way in which hese g oups di e . The coe icien s (˜a) o ICA componen s
will cons i u e he ea u e ec o used o he classi ica ion s ep, as made in Epi anio [38] o uni a ia e unc ions and
Epi anio and Ven u a-Campos [25] o mul i a ia e unc ions wi h one a gumen . The sco es o unc ional PCs can also
be used, al hough in Epi anio and Ven u a-Campos [25] he esul s we e no so good as o ICA. The use o PCA is qui e
common be o e he classi ie is applied, such as in Beg e al. [18] o Shen e al. [13] (al hough hey did no applied PCA
on he coe icien s bu on he landma ks: poin s es ima ed on he su aces).
We p opose o compu e a linea disc iminan ec o unc ion λj(θ, ϕ)=(λj
X(θ, ϕ), λj
Y(θ, ϕ), λj
Z(θ, ϕ)) based on FICA
as done in Ramsay and Sil e man [22, Chap e 8] wi h FPCA. This unc ion λj(θ, ϕ)would be he unc ional coun e pa
o he linea disc iminan o canonical a ia e [47, Chap e 3], he e o e, dj
i=< Fi, λj>=RS2xiλj
XdΩ + RS2yiλj
YdΩ +
RS2ziλj
ZdΩwould e u n he sco e o disc iminan alue o Fi. I we exp ess bo h unc ions in he sphe ical ha monics
base, due o he o hono mali y, dj
iis jus he inne p oduc o wo ec o s, dj
i=(λj)(ci)′, whe e λjis he ec o wi h he
coe icien s o λj(θ, ϕ)in ha base. In his way, he p oblem is educed o ind hese coe icien s in he sphe ical ha monic
expansion.
Assume ha he e a e Qg oups, each o hem wi h size Ni(PQ
i=1 Ni=N) and we apply he s anda d linea
disc iminan analysis (LDA) o he K×Nma ix Awi h he coe icien s o he KICA componen s. This yields a K×
ma ix L( = min{K, Q −1}is he numbe o disc iminan unc ions) gi ing he ×Nma ix Do disc iminan alues
(D=L′A). By equa ion7,A=(S′
bSb)−1S′
bC′, whe e Sbis he 3M×Kma ix con aining he independen componen s
o C′, he N×3Mma ix wi h he coe icien s in he sphe ical ha monics base wi h L= 15 (hence M= 256). As we had
D=ΛC′whe e Λis he ×3Mma ix wi h he coe icien s o he unc ions λj(θ, ϕ)(j= 1, ..., ) in he sphe ical
ha monic base (λjis he j- h ow), hen Λ=L′(S′
bSb)−1S′
b.
As he p oblem has been educed o a MDA p oblem (al hough he basis choice plays a key ole), we can conside
signi icance es s unde he assump ion o mul i a ia e no mali y o he coe icien s in A[48, Sec. 8.6] (ICA looks o
non-gaussiani y in Sno in A).
2.7. Visualiza ion o he esul s
In o de o display he e ec o each unc ional PC, FICA componen o disc iminan unc ion, a small se o sui able
mul iples (posi i e o nega i e) o he unc ion in ques ion is added o he he mean unc ion (mean hippocampus), which
can be displayed o each mul iple sepa a ely. This p ocedu e is usual in shape and FDA li e a u e [22]. Fu he mo e, a
ec o map can be plo ed: ec o s can be d awn om he mean shape o he su ace o med by he mean plus he mul iple
o he unc ion in ques ion. We can also colo he mean hippocampus using he magni ude (no m) o hose ec o s.
Coe icien s in FICA o in FPCA base (sco es) and he disc iminan alues could be also plo ed.
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3. Resul s
The main code (mos ly in Ma lab) and da a a e a ailable a h p://www3.uji.es/∼epi anio/RESEARCH/alz da. a .
Two aluable packages a e: he Su S a package (h p://www.ma h.mcgill.ca/kei h/su s a ) and i s ex ension
(h p://www.s a .wisc.edu/∼mchung/ esea ch/amygdala/) [17] and he Fas ICA package [49]. The Fas ICA algo i hm
(which includes he PCA compu a ion) is used o oba aining ICA. Al hough we ha e no conside ed o he algo i hms
o ob aining ICA, he e is no es ic ion o using any o he algo i hm, bu Fas ICA is an e icien and popula algo i hm.
I is based on a ixed-poin i e a ion scheme maximizing non-Gaussiani y as a measu e o s a is ical independence. We
ha e used he de aul pa ame e s o Fas ICA o ob aining ICA: he independen componen s a e es ima ed one-by-one
(no in pa allel), he nonlinea i y used in he ixed-poin algo i hm is a Gaussian unc ion, and he de aul pa ame e s o
con olling he con e gence (see he so wa e [49] o de ails).
We ha e only conside ed he le hippocampi as in Beg e al. [18] o illus a ing and assessing he p oposed
me hodology. The da abase is qui e small o ob aining alid medical conclusions, al hough he me hodology could be
used wi hou modi ica ion wi h a la ge da abase o he le and igh hippocampi. The le hippocampal olume has been
shown o be be e a disc imina ing MCI s a us [50]. The olume can be es ima ed as he sum o he slice a eas, i.e.
he numbe o pixels belonging o each segmen ed hippocampal slice. Ne e heless, he nume ical esul s o he igh
hippocampi a e shown in he supplemen a y ma e ial.
3.1. Func ional app oach: FPCA and FICA
FPCA is ca ied ou o desc ibe he a iabili y. The i s h ee p incipal componen s explain 49.48% o he whole
a iance, made up o 24.73%, 14.15% and 10.60% espec i ely. 91.27% o he a iabili y is explained by he i s six een
componen s. Figu e 3p esen s isual ep esen a ions o he shape a ia ion along he i s h ee p incipal componen s.
Figu es in he supplemen a y ma e ial can help o in e p e hem. The i s componen co espond o a size componen ,
mos ly concen a ed on he head and he ail, bu also in he body. 45.25% o he a ia ion in his componen is due o he
x-coo dina es (27.17% and 27.58% due o he yand z-coo dina es). Componen 2 is ocused on a pa o he ail. In his
componen , 20.37% o a iabili y comes om he y-coo dina es, and he es is di ided be ween he xand z-coo dina es.
Finally, he hi d componen is concen a ed almos en i ely on he whole ail. The p opo ion o he a iabili y in his
componen is 41.81% o he y-coo dina es (23.51% and 34.67% due o he xand z-coo dina es).
I is in e es ing o dis inguish be ween pa ien s wi h AD, MCI and elde ly CN subjec s, howe e in he li e a u e i
is qui e common o conside he pai -wise compa isons among he h ee g oups. The impo an CN-MCI subp oblem
is analyzed he e, oge he wi h he nume ical esul s o he h ee g oup p oblem. Fo a de ailed analysis o he o he
p oblems, see he supplemen a y ma e ial.
In o de o es he abili y o classi y he subjec s in o hei co ec g oup, c oss- alida ion is pe o med using lea e-
one-ou ials. In each ial, one subjec is se aside and FPCA is pe o med on he emaining subjec s ( he aining se ).
On he one hand, he LDA classi ie is ained wi h he i s JPC sco es o he aining se . On he o he hand, we
compu e (and eco d) he lea e-one-ou (LOU) p edic ion e o o ha aining se . The es subjec is p ojec ed on o he
same p incipal componen s ob ained om aining se alone, and classi ied wi h he ained LDA. This p edic ed class is
p ese ed o p oduce he LOU es ima e o he co ec classi ica ion pe cen age o Jcomponen s, since his p ocess is
epea ed in u n o each o he subjec s. No e ha as FPCA is compu ed wi h di e en da a, PCs om di e en i e a ions
a e no compa able, especially hose wi h low a iance. We ha e he es ima ed accu acy o a ious alues o he numbe
Jo p incipal componen s. In o de o selec he app op ia e numbe o componen s, we do no choose he bes esul ,
as he accu acy would be oo op imis ic, bu a double o nes ed c oss- alida ion is done. We conside he eco ded LOU
p edic ion e o s o each aining se , and build a ma ix wi h hem. The numbe o ows is he numbe o subjec s whe eas
he numbe o columns is he o al numbe o alues Jconside ed. We compu e he mean o each column, and he model
( alue J) selec ed is ha one which gi es he smalles mean (in case o ie, we selec ha wi h less componen s, he mos
pa simonious). Hence, he es ima ed classi ica ion accu acy is he es ima ed accu acy by LOU o he model selec ed by
he nes ed LOU. Table 2shows he esul s o FPCA oge he wi h he es o me hods o he CN s MCI p oblem. In he
supplemen a y ma e ial a able wi h he mean and s anda d de ia ion o each column o he ma ix can be seen, and also
a able wi h he ex e nal LOU accu acies o di e en J alues.
The same c oss- alida ion s a egy conside ed o FPCA is ollowed wi h FICA, whose esul s a e displayed in Table
2. Fu he mo e, we ha e compu ed he unc ional linea disc iminan using all he subjec s in he se o he numbe o
componen sselec ed (J= 6 in hissubp oblem). In Figu e4i is isualized as explainedin Sec. 2.7. I sugges s a small loss
in he CA1 and a pa o he subiculum in he body o he hippocampus. The disc iminan unc ion signi ican ly sepa a es
he g oups (Wilks’ Λ= 0.197, p- alue = 0.002).
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3.2. Compa a i e pe o mance
We ep oduce he me hodology in Chung e al. [17] wi h ou da a. We pe o m mul i a ia e linea modeling [51] on ou
sphe ical ha monic ep esen a ion, es ing he e ec o g oup a iable in he model. The e is no s a is ically signi ican
shape di e ence a α=0.05 when we es o g oup di e ences a each e ex o he hippocampal su ace (see Figu e 5
displaying he F-s a is ic alue on he mean hippocampus). Al hough es ing o g oup mean di e ence and disc iminan
analysis a e di e en p oblems (no e ha he coe icien s o he disc iminan unc ion in MDA a e de i ed so as o
maximize he di e ences be ween he g oup means), we ha e included his esul o highligh ing he u ili y o he
p oposed disc iminan unc ion since signi icance maps o g oup di e ences usually appea in neu oimaging li e a u e
oge he wi h classi ica ion esul s [16].
The me hodology in Ge a din e al. [16] is applied o ou da a: he SPHARM coe icien s a e classi ied wi h a suppo
ec o machine (SVM). S uden ’s - es s we e used o de e mining which coe icien s bes sepa a e he g oups, wi h a
bagging s a egy (we use he same s a egy bu wi h he absolu e alue o he s a is ic o eally keeping only hose
coe icien s which a e always signi ican ly di e en since i is a wo ailed es ). The esul ob ained wi h LOU is in Table
2. The numbe o coe icien s is selec ed by nes ed LOU. A linea ke nel is conside ed, since be e esul s a e ob ained
han wi h adial basis unc ions wi h di e en scaling ac o s.
Recen ly, Clemmensen e al. [52] ha e p oposed a spa se disc iminan analysis (SDA) o he high-dimensional se ing
( he numbe o a iables is la ge ela i e o he numbe o subjec s). This me hod pe o ms linea disc iminan analysis
wi h a spa seness c i e ion imposed such ha classi ica ion and ea u e selec ion a e pe o med simul aneously. SDA is
applied o ou SPHARM coe icien s. We use he spa seLDA package wi h de aul pa ame e s (excep o he desi ed
numbe o coe icien s o be selec ed), which is a ailable om h p://www2.imm.d u.dk/∼lhc/. The esul ob ained wi h
LOU is in Table 2. The numbe o coe icien s is chosen by nes ed LOU.
Beg e al. [18] p oposed ou shape ea u es o disc imina ing CN s MCI wi h le hippocampi. They epo ed he
ollowing accu acies o each se o ea u es: 68.1% o 3D momen in a ian s ea u es, 75% o 3D enso in a ian
ea u es, 77.3% o 3D Laplacian in a ian ea u es, and 86.3% o 3D geodesic shape in a ian s ea u es. No e ha in
hei da abase he e a e 26 CN and 18 MCI (a p opo ion simila o ou da abase). Al hough esul s should be compa ed
wi h cau ion since he da abases a e di e en , FPCA and FICA ob ain be e accu acies.
O he app oaches, such as ha in Shen e al. [13], e alua e he sphe ical ep esen a ion in a se ie o pa ame e loca ions,
ob aining he coo dina es on he su ace, and wo k wi h hese landma ks (SPHARM-PDM om Poin Dis ibu ion Model)
o classi ica ion. We do no conside his app oach since i is well-known ha some egula iza ion is necessa y in o de o
ob ain meaning ul esul s [19, Ch. 11], [22, Ch. 8]. In ac , in Figu e 5, whe e each spa ial loca ionis conside ed sepa a ely,
he e a e a lo o small spo s. Howe e , in Figu e 4 he disc imina i e zones a e la ge and mo e homogeneous.
Table 3shows he con usion ma ices o each me hod when he h ee g oups a e conside ed join ly, excep o SVM in
Ge a din e al. [16], whose me hodology is only o p oblems wi h wo g oups. Bes o equal accu acies a e achie ed wi h
he unc ional app oach. The CN g oup is co ec ly iden i ied by FPCA, FICA and SDA, and SDA classi ies co ec ly he
AD g oup ( he e is only one misclassi ica ion is his g oup o he es o me hods, which classi ied an AD indi idual as
MCI). The main e o s a e in he MCI g oup: wi h olumen only one indi idual is co ec ly iden i ied in his g oup ( wo
indi iduals a e misclassi ied as CN and h ee as AD); wi h SDA wo indi iduals a e misclassi ied as CN and o he wo as
AD, while he less numbe o e o s a e achie ed wi h FPCA and FICA, whe e wo indi iduals a e misclassi ied as CN
and one as AD. The wo MCI indi iduals misclassi ied as CN a e he same o all he me hods, as well as he misclassi ied
MCI indi idual as AD o FPCA and FICA.
4. Conclusions
Ou no el con ibu ion is o model he 3D hippocampal su aces wi h a FDA app oach, as hippocampi a e in ac unc ions
de ined o e wo sphe ical angles. This app oach allows o ca y ou he same analysis as any o he b anch o s a is ics
[19]. As he e a e ew s udies wi h unc ional da a wi h mo e han one a gumen , we ha e ex ended he unc ional da a
me hodology o unc ions wi h wo a gumen s (angles in his case), in oducing FICA o he i s ime. FPCA and FICA
wi h wo a gumen s ha e been used o shape desc ip ion and classi ica ion. A unc ional linea disc iminan wi h FICA
has been also de ined. To he bes o ou knowledge i has been used o he i s ime in localizing he di e ences among
g oups ins ead o he signi icance maps, and meaning ul esul s a e ob ained wi h i . The unc ional linea disc iminan
wi h FICA has gi en mo e de ails and cohe ence han ha wi h FPCA (see he supplemen a y ma e ial). Classi ica ion
esul s a e be e wi h FPCA and FICA (bo h wi h iden ical esul s), han he o he al e na i es conside ed, showing ha
ea u e ex ac ion is a powe ul me hod [53, Sec. 5.3.]. The same conclusions a e eached wi h he o he subp oblem
conside ed in supplemen a y ma e ial. The good esul s in his impo an p oblem a e o he o ou main con ibu ions:
in Table 2highe accu acy, sensi i i y and speci ici y is achie ed wi h he FDA app oach (FPCA and FICA) han wi h
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Recei ed XXXX
(www.in e science.wiley.com) DOI: 10.1002/sim.0000
Supplemen a y ma e ial o “Hippocampal shape
analysis in Alzheime ’s disease using Func ional
Da a Analysis”
I ene Epi anioa∗, Noelia Ven u a-Camposb
1. In oduc ion
This Supplemen a y Ma e ial con ains he analysis o he AD pa ien s e sus CN and e sus MCI and he analysis o he
h ee g oups: CN, MCI and pa ien s wi h ea ly AD.
2. Resul s
Figu e 1shows he magni ude o he i s h ee p incipal componen o e he mean hippocampus o he whole da abase.
The iewpoin s ha e been selec ed in o de o isually app ecia e be e he e ec . As code and da a a e a ailable a
h p://www3.uji.es/∼epi anio/RESEARCH/alz da. a , igu es can be ep oduced and he iew can be in e ac i ely o a ed.
Head, body and ail a e he h ee pa s ha make up a hippocampus [1]. A schema ic ep esen a ion o he hippocampal
sub ields is shown in Figu e 2, which can help in he in e p e a ion. Analogously, Figu e 3shows he magni ude o he
i s nine independen componen s o e he mean hippocampus o he whole da abase. The e ec o he i s and hi d
componen s is dis ibu ed along he whole hippocampus excep he mo e ex eme zones o he ail and head; while he es
o componen s a e concen a ed in di e en pa s o he hippocampus: ail (second, se en h, eigh h and nin h), subiculum
( i h and six h), and head ( ou h).
Table 1gi es he accu acies o he CN s MCI p oblem using di e en alues o L o ep esen ing he hippocampi.
No e ha he esul s a e simila o hose ob ained wi h L=15, bu when Linc eases ( o Lbigge han he chosen L=
15) he pe o mances a e a bi wo se, maybe because we could be i ing noise. Fo small L he pe o mances a e simila .
Cu iously wi h L= 3 he nume ical esul s a e a bi be e , bu i would no be possible o app ecia e whe e he di e ences
a e wi h he unc ional linea disc iminan , since wi h L= 3 he deg ee o smoo hing is e y high (see Figu e 2 in he
pape ).
Table 2gi es he summa y analysis o he accu acies o he nes ed LOU ( he eco ded LOU p edic ion e o s o
each aining se ) o di e en alues o Jin he CN s MCI subp oblem. The bigges mean accu acy o each me hod
co esponds wi h one o he smalles s anda d de ia ion. No e ha hese accu acies a e bigge han hose epo ed in
Table 2 o he pape , since hese accu acies a e compu ed wi h each aining se , and he e o e hey a e o e es ima ed
some imes subs an ially [2, ch. 7], while in Table 2 o he pape he ex e nal LOU esul s a e epo ed (each es subjec is
a comple ely new da a no used in he analysis o he selec ion o he model, o asce aining he gene aliza ion capabili y
o he classi ie ). In Table 3 he accu acies o he ex e nal LOU o di e en alues o Ja e shown ( he selec ed ones by
he nes ed LOU, whose pe o mances appea in Table 2 o he pape , a e in a ame box).
aDep . Ma em`a iques, Uni e si a Jaume I, Campus del Riu Sec, 12071 Cas ell´o, Spain
bDep . Psicologia B`asica, Cl´ınica i Psicobiologia, Uni e si a Jaume I, Spain
∗Co espondence o: Tel.: +34-964728390, ax: +34-964728429. E-mail: epi [email protected]
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Figu es 4,5and 6display wo iews o he unc ional linea disc iminan wi h FICA using all he subjec s in each
subse o he numbe o componen s selec ed by nes ed LOU. Conside ing all hese igu es join ly, oge he wi h Figu e
3 in he pape , he e olu ion o he disease (whe e i begins and whe e i ends) can be seen pe ec ly. As said in he pape ,
i sugges s a small loss in he CA1 and a pa o he subiculum in he body o he hippocampus o CN s MCI. In MCI
s AD, he e is a la ge di e ence in CA1, wi h a bigge alue in he ail. Hence, he loss would begin om he head o
he ail. Fu he mo e, he e a e many di e ences along he whole subiculum, no so localized as in he CN s MCI case. In
summa y, i is clea ha he disease would begin in he CA1 egion om he head o he ail, wi h a p og essi e loss in he
subiculum, al hough his loss is no so p onounced as in CA1.
Figu e 7shows he unc ional linea disc iminan wi h FPCA de ined by [3, ch. 8] o he di e en subp oblems. The
iewpoin s ha e been selec ed in o de o isually app ecia e be e he e ec . No e ha he disc imina ing abili y o
he disc iminan unc ions in he MCI s AD and CN s AD subp oblems is in doub since hei espec i e p- alues a e
highe han α= 0.05. When he h ee g oups a e conside ed join ly, we ob ain he esul s expec ed in he hippocampal
head (CA1 sub egion), and he ail o he hippocampus wi h a simila le el o disc imina ion, and a loss in he subiculum
in he hippocampal body, al hough no so p onounced. As ega ds he subp oblems CN s MCI and CN s AD, we
ob ain he a eas ha we e expec ed: he head he hippocampus (CA1 sub egion) and he ail o he hippocampus, wi h
g ea e disc imina ion in he CN s MCI subp oblem. This is no cohe en since he AD pa ien s ha e a g ea e a ophy o
hippocampal olume loss, so i would be expec ed a highe alue o disc imina ion be ween CN s AD. Fo he MCI s
AD subp oblem, he same con igu a ion (disc iminan egions) ha o FICA is ob ained. Howe e , he disc iminan le el
is equal o he mo e disc iminan zones (CA1 sub egion in he on o he head and he ail o he hippocampus), whe eas
wi h FICA i can be obse ed no only a high disc imina ion in he CA1 sub egion loca ed in he on o he head, bu
also along he body o he hippocampus and a bigge di e ence in he ail o he hippocampus. The analysis wi h FPCA
shows a clea disad an age wi h ha done wi h FICA because based on i we could de e mine he p og ess o he disease,
whe eas wi h FPCA he e is no a p og essi e change in h esholding (colo ), i.e. i ma ks s ongly whe e he di e ences
a e, bu FICA gi es g ea e de ails.
Tables 4and 5gi e he pe o mance o he espec i e subp oblem. Bes o equal accu acies a e achie ed wi h he
unc ional app oach. No e ha 100% co ec classi ica ions o CN s AD a e ob ained by all me hods excep o SDA,
bu SVM uses 20 ea u es, when he o al numbe o subjec s in ha subp oblem is 22.
I could be in e es ing o plo he sco es o each subjec on di e en componen s because hese sca e plo s can e eal
in e es ing ea u es, such as he dis ibu ion o he subjec s on hose componen s, clus e s o subjec s, ou lie s, e c. [4].
No e ha he complex in o ma ion in he hippocampi, which a e s uc u es in 3D, will be ep esen ed wi h simple sca e
plo s. Fo he MCI s AD subp oblem, which is he mos di icul subp oblem acco ding o he ob ained accu acies,
we ha e compu ed he sco es ( he ea u es used wi h LDA) using all he subjec s in ha subse (16) o he numbe o
componen s selec ed (J= 3 o FPCA and J= 4 o FICA in his subp oblem). In Figu e 8 hose sco es o FPCA and
FICA o he wo componen s ha isually bes e lec he sepa a ion be ween g oups a e ep esen ed. The sca e plo o
FICA shows a sligh g ea e sepa a ion han ha o FPCA, wi h wo pa ien s wi h di e en condi ions nea ly o e lapped.
Ins ead o he sco es used in he classi ica ion, in Figu e 9we show he disc iminan alues o he wo disc iminan
unc ions o he h ee g oups join ly, using all he subjec s in ha se (28) o he numbe o componen s selec ed (J= 8
o FPCA and J= 9 o FICA in his p oblem). C osses, s a s and ci cles ep esen he CN, MCI, and pa ien s wi h ea ly
AD, espec i ely. The plo s o FPCA and FICA a e nea ly iden ical. One o he MCI pa ien ( he same in bo h plo s) is
nea he AD g oup.
Table 6shows he accu acies o each subp oblem when he igh hippocampi a e analyzed ( he esul s o he le
hippocampi a e also shown in o de o make easie he compa isons). As in p e ious wo ks [5,6,7,8,9,10,11,12,13,
14,15,16], i seems ha he le hippocampi can disc imina e be e he AD condi ion han he igh hippocampi.
Re e ences
[1] Hasboun, D., Chan ˆome, M., Zouaoui, A., Sahel, M., Deladoeuille, M., Sou ou , N., Duyme, M., Baulac, M.,
Ma saul , C., and Do mon , D. MR de e mina ion o hippocampal olume: Compa ison o h ee me hods. Ame ican
Jou nal o Neu o adioly, 17:1091–1098, 1996.
[2] Has ie, T., Tibshi ani, R., and F iedman, J. The Elemen s o S a is ical Lea ning. Da a mining, in e ence and
p edic ion. Sp inge -Ve lag, second edi ion, 2009.
[3] Ramsay, J. O. and Sil e man, B. W. Applied Func ional Da a Analysis. Sp inge , 2002.
[4] Jolli e, I. T. P incipal Componen Analysis. Sp inge , second edi ion, 2002.
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[5] Beg, M. F., Raamana, P. R., Ba bie i, S., and Wang, L. Compa ison o ou shape ea u es o de ec ing hippocampal
shape changes in ea ly Alzheime s. S a is ical Me hods in Medical Resea ch, published online 30 May 2012(DOI:
10.1177/0962280212448975):1–24,2012.
[6] M¨ulle , M. J., G e e us, D., Weib ich, C., Dellani, P. R., Scheu ich, A., S oe e , P., and Fellgiebel, A. Diagnos ic
u ili y o hippocampal size and mean di usi i y in amnes ic MCI. Neu obiology o Aging, 28(3):398–403, 2007.
[7] Epi anio, I. and Ven u a-Campos, N. Func ional da a analysis in shape analysis. Compu a ional S a is ics & Da a
Analysis, 55(9):2758–2773, 2011.
[8] Wang, L., Beg, M. F., Ra nana he , J. T., Ce i oglu, C., Younes, L., Mo is, J. C., Cse nansky, J. G., and Mille ,
M. I. La ge de o ma ion di eomo phism and momen um based hippocampal shape disc imina ion in demen ia o
he Alzheime ype. IEEE T ans. Med. Imaging, 26(4):462–470, 2007.
[9] Ecke s ¨om, C., Olsson, E., Bo ga, M., Ekholm, S., Ribbelin, S., Rols ad, S., S a ck, G., Edman, ˚
A., Wallin, A., and
Malmg en, H. Small baseline olume o le hippocampus is associa ed wi h subsequen con e sion o MCI in o
demen ia. The G¨o ebo g MCI s udy. Jou nal o he Neu ological Sciences, 272:48–49, 2008.
[10] Wol , H., G unwald, M., K uggel, F., Riedel-Helle , S. G., Ange h¨o e , S., Hojja oleslami, A., Hensel, A., A end ,
T., and Ge z, H. Hippocampal olume disc imina es be ween no mal cogni ion; ques ionable and mild demen ia in
he elde ly. Neu obiol Aging, 22(2):177–186, 2001.
[11] Woola d, A. and Hecke s, S. Ana omical and unc ional co ela es o human hippocampal olume asymme y.
Psychia y Res, 201(1):48–53, 2012.
[12] Vijayakuma , A. and Vijayakuma , A. Compa ison o hippocampal olume in demen ia sub ypes. ISRN Radiology,
A icle ID 174524(doi:10.5402/2013/174524):5,2013.
[13] Ca ne, R. P., Vog in, S., Li ewka, L., and Cook, M. J. Ce eb al co ex: an MRI-based s udy o olume and a iance
wi h age and sex. J Clin Neu osci, 13(1):60–72, 2006.
[14] Fox, N. C., Wa ing on, E. K., F eebo ough, P. A., Ha ikainen, P., Kennedy, A. M., S e ens, J. M., and Rosso ,
M. N. P esymp oma ic hippocampal a ophy in Alzheime ’s disease. a longi udinal MRI s udy. B ain, 119(6):
2001–7, 1996.
[15] Laakso, M. P., Pa anen, K., J , Leh o i a, M., Helkala, E. L., Hallikainen,M., Hanninen, T., Vainio,P., and Soininen,
H. Hippocampal olumes in Alzheime ’s disease, Pa kinson’s disease wi h and wi hou demen ia, and in ascula
demen ia: An MRI s udy. Neu ology, 46(3):678–681, 1996.
[16] K asuski, J., Alexande , G., Ho wi z, B., Daly, E., Mu phy, D., Rapopo , S., and Schapi o, M. Volumes o
medial empo al lobe s uc u es in pa ien s wi h Alzheime ’s disease and mild cogni i e impai men (and in heal hy
con ols). Biol Psychia y, 43(1):60–8, 1998.
[17] Ge a din, E., Che ela , G., Chupin, M., Cuingne , R., Desg anges, B., Kim, H., Nie hamme , M., Dubois, B.,
Lehe icy, S., Ga ne o, L., Eus ache, F., and Collio , O. Mul idimensional classi ica ion o hippocampal shape
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(4):1476–1486, 2009.
3. Figu es and Tables
Table 1. Accu acies (%) o he CN s MCI pa ien s o he di e en alues o L( he numbe o ea u es be ween
pa en heses), selec ing wi h double lea e-one-ou he ea u es and using hese ea u es wi h lea e-one-ou o ob aining
he accu acy.
Me hod L= 2 L= 3 L= 5 L= 7 L= 9 L= 11 L= 13 L= 15 L= 17 L= 19 L= 21
FPCA 88.89 (4) 94.44 (4) 88.89 (4) 88.89 (4) 88.89 (4) 83.33 (4) 88.89 (5) 88.89 (5) 83.33 (4) 77.78 (4) 77.78 (4)
FICA 88.89 (5) 94.44 (5) 88.89 (4) 83.33 (4) 83.33 (4) 88.89 (6) 88.89 (6) 88.89 (6) 83.33 (5) 83.33 (5) 77.78 (4)
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Figu e 1. The e ec o he i s PC, he second PC and he hi d PC ( om le o igh ) on he mean shape o 2 s anda d de ia ions o each componen .
Figu e 2. Zones on he le hippocampal su ace. CA1 is in ed, subiculum in yellow, and CA2, CA3, CA4 and gy us den a us in blue.
Table 2. Mean accu acies and hei espec i e s anda d de ia ions be ween pa en heses o he nes ed LOU, o he CN s
MCI pa ien s, o di e en alues o J.
Me hod J= 1 J= 2 J= 3 J= 4 J= 5 J= 6 J= 7 J= 8 J= 9 J= 10 J= 11 J= 12 J= 13
FPCA 72.88 (5.93) 71.24 (4.34) 82.35 (6.50) 91.18 (4.90) 92.48 (3.83) 86.93 (5.39) 83.66 (5.39) 83.99 (4.72) 80.39 (3.92) 76.14 (5.71) 74.18 (7.11) 73.86 (10.04) 74.51 (6.20)
FICA 60.78 (8.08) 69.28 (4.62) 72.55 (3.40) 87.91 (8.87) 87.58 (5.84) 88.56 (4.15) 85.95 (4.44) 80.07 (5.24) 78.76 (5.59) 73.53 (6.86) 67.65 (7.40) 66.01 (6.95) 66.34 (6.74)
SVM [17] 69.28 (8.45) 69.28 (8.67) 70.26 (9.09) 78.10 (6.74) 69.28 (10.84) 69.28 (9.71) 80.72 (6.73) 78.43 (8.77) 94.12 (3.40) 94.12 (3.40) 93.46 (3.33) 95.10 (2.94) 94.12 (3.40)
SDA 84.97 (4.48) 86.60 (6.14) 89.22 (6.58) 92.48 (6.45) 95.42 (5.02) 98.04 (3.40) 99.35 (2.69) 99.35 (2.69) 99.35 (2.69) 99.67 (1.35) 100 (0) 100 (0) 100 (0)
Table 3. Accu acies o he ex e nal LOU, o he CN s MCI pa ien s, o di e en alues o J.
Me hod J= 1 J= 2 J= 3 J= 4 J= 5 J= 6 J= 7 J= 8 J= 9 J= 10 J= 11 J= 12 J= 13
FPCA 72.22 66.67 72.22 83.33 88.89 83.33 83.33 88.89 83.33 88.89 88.89 83.33 83.33
FICA 61.11 66.67 72.22 77.78 83.33 88.89 83.33 83.33 83.33 83.33 88.89 88.89 77.78
SVM [17] 61.11 61.11 61.11 61.11 50 50 50 50 61.11 61.11 50 77.78 61.11
SDA 72.22 72.22 72.22 72.22 72.22 72.22 72.22 72.22 72.22 72.22 72.22 77.78 77.78
Table 4. Pe o mance o he MCI s AD pa ien s o he di e en me hods, selec ing wi h double lea e-one-ou he
ea u es and using hese ea u es wi h lea e-one-ou o ob aining he accu acy.
Me hod Volume FPCA FICA SVM [17] SDA
Accu acy (%) 75.00 87.50 87.50 68.75 62.5
No. ea u es 1 3 4 8 6
Sensi i i y (%) 80.00 100 100 90.00 80.00
Speci ici y (%) 66.67 66.67 66.67 33.33 33.33
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Figu e 3. The e ec o he i s nine independen componen s on he mean shape.
Figu e 4. Func ional linea disc iminan wi h FICA o MCI s AD (J= 4 componen s). The disc iminan unc ion signi ican ly sepa a es he g oups (Wilks’ Λ= 0.320, p- alue
= 0.008).
Table 5. Pe o mance o he CN s AD o he di e en me hods, selec ing wi h double lea e-one-ou he ea u es and
using hese ea u es wi h lea e-one-ou o ob aining he accu acy.
Me hod Volume FPCA FICA SVM [17] SDA
Accu acy (%) 100 100 100 100 90.91
No. ea u es 1 1 2 20 6
Sensi i i y (%) 100 100 100 100 90.00
Speci ici y (%) 100 100 100 100 91.67
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Figu e 5. Func ional linea disc iminan wi h FICA o CN s AD (J= 2 componen s). The disc iminan unc ion signi ican ly sepa a es he g oups (Wilks’ Λ= 0.103, p- alue =
0).
Figu e 6. Fi s unc ional linea disc iminan wi h FICA o CN, MCI and AD (J= 9 componen s). The wo disc iminan unc ions signi ican ly sepa a e he g oups a α= 0.05,
bu only he i s one, whose p opo ion o ace is 89.29%, is displayed ( i s : Wilks’ Λ= 0.046, p- alue = 0; second: Wilks’ Λ= 0.474, p- alue = 0.048).
Table 6. Accu acies (%) o he di e en subp oblems o bo h hippocampi ( igh /le ), selec ing wi h double lea e-one-
ou he ea u es and using hese ea u es wi h lea e-one-ou o ob aining he accu acy. The numbe o ea u es is be ween
pa en heses.
Me hod 3 g oups CN s MCI MCI s AD CN s AD
FPCA 75 (1) / 85.71 (8) 72.22 (3) / 88.89 (5) 81.25 (5) / 87.5 (3) 100 (1) / 100 (1)
FICA 85.71 (4) / 85.71 (9) 66.67 (4) / 88.89 (6) 81.25 (4) / 87.5 (4) 95.45 (2) / 100 (2)
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Figu e 7. Top le : Fi s unc ional linea disc iminan wi h FPCA o CN, MCI and AD (J= 8 componen s), whose p opo ion o ace is 92.9% and Wilks’ Λ= 0.065, p- alue
= 0 (no e ha o he second unc ional linea disc iminan he p- alue is highe han α= 0.05, Wilks’ Λ= 0.609, p- alue = 0.154); Top igh : Func ional linea disc iminan wi h
FPCA o CN s MCI (J= 5 componen s, Wilks’ Λ= 0.266, p- alue = 0.003); Bo om le : Func ional linea disc iminan wi h FPCA o MCI s AD (J= 3 componen s, Wilks’
Λ= 0.612, p- alue = 0.105, no e ha he p- alue is highe han α= 0.05); Bo om igh : Func ional linea disc iminan wi h FPCA o CN s AD (J= 1 componen , Wilks’ Λ=
0.959, p- alue = 0.367, no e ha he p- alue is highe han α= 0.05).
−5 0 5 10
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Figu e 8. Sca e plo o sco es o MCI s AD. Componen 1 s 3 o FPCA (le ) and componen 2 s 4 o FICA ( igh ). S a s and ci cles ep esen he MCI and pa ien s wi h
ea ly AD, espec i ely.
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26 28 30 32 34 36 38
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Figu e 9. Sca e plo o disc iminan alues o he wo disc iminan unc ions o he h ee g oups: FPCA (le ) and FICA ( igh ). C osses, s a s and ci cles ep esen he CN, MCI,
and pa ien s wi h ea ly AD, espec i ely.
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